To complement the “conceptual” images we created to depict the Prisoner’s Dilemma, Hawk-Dove, and Stag Hunt games, Greg Riestenberg and I have been developing a series of “example” images showing how the payoffs of these games are produced by a variety of classic “realistic” examples. The goal with these images is to provide teachers and learners with simple, easy-to-visualize examples of how two players might interact in these different game scenarios. Whereas the conceptual images use abstract symbols to represent variable payoffs, these matrix versions of each game show actual payoffs. To provide continuity with the conceptual graphics and communicate which game scenario is depicted, each image features the “brand image” for the Prisoner’s Dilemma, Hawk-Dove, and Stag Hunt game in the upper-left corner.

The first image is inspired by Douglas Hofstader’s “closed bag exchange” version of the Prisoner’s Dilemma:

In this image, we depict a buyer and a seller exchanging closed briefcases, each of which could either contain the goods and payment agreed upon for exchange or nothing. Providing the opposing player with either goods or payment is to cooperate, whereas providing an empty briefcase is to defect. The image represents the outcome for each player in all four possible combinations of game play.

Our second Prisoner’s Dilemma image depicts a “cold war” military scenario where two countries labeled “A” and “B” must decide whether to invest in an offensive military:

We use flags to represent each country: the flags are differentiated by color, pattern, and a label letter. Under this scenario cooperation translates to not wasting your resources on an offensive military, while defection translates to spending resources on an offensive military. If both countries cooperate, they coexist peacefully and maintain their individual sovereignty, allowing them to retain resources for more constructive purposes. We represent this outcome by showing each country’s flag unaltered next to its own currency. If one country cooperates while the other defects, the defecting country uses its military might to capture the other country and usurp its resources. We show this conquest by depicting the flag of the defecting country in the color of the cooperating country, suggesting that the price of cooperating is to be dominated by the defecting country. The payoff to the defecting country is represented by the currency of the cooperating country, which is now depicted in the color of the defecting country to show change in ownership. We show why the defecting country is able to usurp these resources by depicting a nuclear warhead aimed at the cooperating country. Under a Prisoner’s Dilemma depiction of military conflict, dual defection is assumed to result in a military standoff. We show this “cold war” outcome by depicting each country maintaining its sovereignty (represented by the flags) but also no longer retaining precious resources, which have been spent to maintain an arsenal of warheads aimed at the other country.

The alternative way to depict military conflict through simple evolutionary games is to use the Hawk-Dove game, as shown in the image below:

This image is very similar to the military version of the Prisoner’s Dilemma, because these two games are very similar. But the Hawk-Dove game differs from the Prisoner’s Dilemma in one important way: in the Hawk-Dove game, dual defection is the worst collective and individual outcome. We show this difference by making a different assumption about the meaning of simultaneous military build-up: rather than simply resulting in a resource-wasting standoff, dual defection results in mutual destruction. Military escalation by each country is assumed to lead to inevitable conflict. We represent the resulting mutual annihilation as two mushroom clouds. Many ecologists have pointed out that the “hawk” and “dove” strategies of this game have no ecological meaning: these are labels used to describe military approaches, not the behavior of these animal species. To underscore this point, we have included a “hawk” icon next to the choice to defect and a “dove” next to the choice to cooperate.

We also provide example images for the Stag Hunt, starting with our best attempt at depicting the original scenario described by Jean-Jacques Rousseau:

In this scenario, it is assumed that two hunters agree to wait behind separate bushes to ambush a stag. If both players wait patiently for the stag to pass by, they ambush the stag successfully, taking home a large parcel of meat: this is considered dual cooperation. We represent the result of this dual cooperation with a large stag shaded in a color that is intermediate to the colors representing each player (showing that the stag is shared). In this version of the Stag Hunt, defection is imagined as giving in the temptation to leave one’s post behind the bush to grab a nearby rabbit. If one player cooperates while his partner defects, he gets nothing: even if the stag passes by, the hunt is unsuccessful because the single cooperating player is unable to bring down this large animal alone. We represent defection by showing the defecting player turned away from the cooperating player, with a single rabbit as his payoff. Dual defection simply results in each player catching his own rabbit.

Another scenario we looked at as being well-represented by the Stag Hunt is the phenomenon of bubble-net feeding by humpback whales. While in reality this feeding strategy relies on the cooperative efforts of many whales, we created an idealized version representing the choices of just two whales:

When humpback whales use bubble-net feeding, they cooperate to corral a school of prey fish in a cylinder of bubbles and then in synchrony dive to the surface to each gain an equal share of prey. By maintaining this cooperation each whale secures the highest possible payoff. We represent this best-possible outcome by showing the whales curved together to corral the entire school of fish, which presumably will be shared equally. Whales can defect on each other if they “break ranks” and dive into the school of prey prematurely. We represent this defection by showing the defecting whale turned away from the cooperating whale, catching a larger haul of fish as a result. Crucially, we show that the breakdown of cooperation allows most of the fish to escape from both whales. If both whales defect, they both catch only a few fish, with even more fish escaping.

We are looking for feedback on these images from anyone with interest and/or expertise in these evolutionary games. Let us know what you think! Feel free to use the “comment” function at the end of this post, to email us directly, or to use my contact form.

Please keep in mind that these images are not yet available for public use. As such we retain the copyright to all images. Please be patient while we complete our refinement process: as soon as we feel these images are complete, they will be made available for public use.